This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.

This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.

The proofs of these properties are both interesting and insightful. They illustrate the power of the third axiom, and its interaction with the remaining two axioms. When studying axiomaticprobability theory, many deep consequences follow from merely these three axioms. In order to verify the monotonicity property, we set and , where for . It is easy to see that the sets are pairwise disjoint and . Hence, we obtain from the third axiom that

Since the left-hand side of this equation is a series of non-negative numbers, and that it converges to which is finite, we obtain both and . The second part of the statement is seen by contradiction: if then the left hand side is not less than

If then we obtain a contradiction, because the sum does not exceed which is finite. Thus, . We have shown as a byproduct of the proof of monotonicity that .

This is called the addition law of probability, or the sum rule. That is, the probability that AorB will happen is the sum of the probabilities that A will happen and that B will happen, minus the probability that both AandB will happen. The proof of this is as follows: